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Working Paper 01-07
Statistics and Econometrics Series 04
February 2001
Departamento de Estadstica y Econometra
Universidad Carlos III de Madrid
Calle Madrid, 126
28903 Getafe (Spain)
Fax (34) 91 624-98-49
OUTLIERS AND CONDITIONAL AUTOREGRESSIVE HETEROSCEDASTICITY
IN TIME SERIES
M. Angeles Carnero, Daniel Pea and Esther Ruiz*
Abstract
This paper reviews the literature on GARCH-type models proposed to represent the dynamic
evolution of conditional variances. Effects of level outliers on the diagnostic and estimation of
GARCH models are also studied. Both outliers and conditional heteroscedasticity can generate
time series with excess kurtosis and autocorrelated squared observations. Consequently, bothphenomena can be confused. However, since outliers are generated by unexpected events and the
conditional variances are predictable, it is important to identify which one is producing the
observed features in the data. We compare two alternative procedures for dealing with the
simultaneous presence of outliers and conditional heteroscedasticity in time series. The first one
is to clean the series of outliers before fitting a GARCH model. The second is to estimate first the
GARCH model and then to clean of outliers by using the residuals adjusted by its conditional
variance. It is shown that both approaches may result in different estimated conditional variances.
Keywords: GARCH; EGARCH; CHARMA; Stochastic Volatility; Asymmetry; autocorrelation
of squares; kurtosis; robust procedures.
*Department of Statistics and Econometrics; Universidad Carlos III de Madrid, Getafe (Madrid),
Carnero: e-mail: [email protected]; Pea: e-mail: [email protected]; Ruiz: e-
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0 1000 2000 30008
6
4
2
0
2
4
6
S&P 500
0 500 1000 1500 20006
4
2
0
2
4
US Dollar/Japanese Yen exchange rate
10 5 0 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7Normal and estimated density
densityestimationnormal
6 4 2 0 2 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7Normal and estimated density
densityestimationnormal
5 10 15 200.2
0
0.2
0.4
ACF of the series
5 10 15 200.2
0
0.2
0.4
ACF of the series
5 10 15 200.2
0
0.2
0.4
ACF of the squared observations
5 10 15 200.2
0
0.2
0.4
ACF of the squared observations
5 10 15 200.2
0
0.2
0.4
ACF of the absolute values
5 10 15 200.2
0
0.2
0.4
ACF of the absolute values
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
5
10
15
20
25
30
First order autocorrelation of squares
kurtosis
+=0.99
+=0.95
+=0.95
+=0.99
+=0.99
ARCH
GARCH
Data
GARCHt10
GARCHt7
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0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.5
1
1.5
2
2.5USJA
originalno marginal aono condic. ao
0 200 400 600 800 1000 1200 14001
1.5
2
2.5
3
3.5
4BOMBAY
originalno marginal aono condic. ao
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0 100 200 300 400 500
5
0
5
Normal ARCH(1) with =0.15
0 100 200 300 400 500
5
0
5
Normal ARCH(1) with =0.4
0 100 200 300 400 500
5
0
5
Normal GARCH(1,1) with =0.1
0 100 200 300 400 500
5
0
5
Normal GARCH(1,1) with =0.2
0 100 200 300 400 500
5
0
5
ARCH(1)t7
0 100 200 300 400 500
5
0
5
GARCH(1,1)t7
yt1
t2
GARCH
AVGARCH
EGARCH
0
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
5
10
15
20
25
First order autocorrelation of squares
Kurtosis
=0.99=0.95
+=0.99
+=0.95
GARCH
EGARCH
Data
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0 0.2 0.40
0.1
0.2
0.3
0.4
=0.1
corrected
0 0.1 0.20
0.05
0.1
0.15
0.2
=0.1
0.6 0.8 10.5
0.6
0.7
0.8
0.9
=0.8
0 0.2 0.40
0.5
1
3consecutiveLO
0 0.2 0.40
0.2
0.4
0.6
0 0.5 10
0.5
1
0 0.5 10
0.5
1
3isolatedLO
original0 0.2 0.4
0
0.1
0.2
0.3
0.4
original0 0.5 1
0
0.5
1
original
32
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38
3
8
33
83
17
92
8
14
86
85
15
56
44
68
32
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